Question

A body of mass 2 kg begins to move under the influence of time dependent force \(\vec{F} = (2t \hat{i} + 6t^2 \hat{j})\) N, where \(\hat{i}\) and \(\hat{j}\) are unit vectors along x and y-axis respectively. The power produced by the force at \(t = 2\) s is ______ W.

Answer 1

Correct Answer: 200

Step 1: Understanding the Concept:
Power delivered by a force is the dot product of the force vector and the velocity vector. Since the force is time-dependent, the acceleration and consequently the velocity will also be time-dependent functions found through integration.
Step 2: Key Formula or Approach:
1. Acceleration: \(\vec{a}(t) = \frac{\vec{F}(t)}{m}\)
2. Velocity: \(\vec{v}(t) = \int \vec{a}(t) dt\) (with initial velocity \(\vec{v}(0) = 0\) as it "begins to move").
3. Power: \(P(t) = \vec{F}(t) \cdot \vec{v}(t)\).
Step 3: Detailed Explanation:
Given:
Mass, \(m = 2 \text{ kg}\).
Force, \(\vec{F}(t) = 2t\hat{i} + 6t^2\hat{j}\).
Find the acceleration vector:
\(\vec{a}(t) = \frac{2t\hat{i} + 6t^2\hat{j}}{2} = t\hat{i} + 3t^2\hat{j} \text{ m/s}^2\).
Find the velocity vector by integrating the acceleration with respect to time:
\(\vec{v}(t) = \int (t\hat{i} + 3t^2\hat{j}) dt = \frac{t^2}{2}\hat{i} + t^3\hat{j} \text{ m/s}\).
(Integration constant is 0 since the body starts from rest at \(t=0\)).
Now calculate the instantaneous power:
\(P(t) = \vec{F}(t) \cdot \vec{v}(t)\)
\(P(t) = (2t\hat{i} + 6t^2\hat{j}) \cdot \left( \frac{t^2}{2}\hat{i} + t^3\hat{j} \right)\)
\(P(t) = (2t)\left(\frac{t^2}{2}\right) + (6t^2)(t^3)\)
\(P(t) = t^3 + 6t^5\).
Evaluate the power at \(t = 2 \text{ s}\):
\(P(2) = (2)^3 + 6(2)^5\)
\(P(2) = 8 + 6(32) = 8 + 192 = 200 \text{ W}\).
Step 4: Final Answer:
The power produced by the force at \(t = 2 \text{ s}\) is 200 W.